Internal problem ID [8529]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 951.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Abel]
\[ \boxed {y^{\prime }+\frac {x}{2}-1-y^{2}-\frac {y x^{2}}{2}-y a x -\frac {x^{4}}{16}-\frac {a \,x^{3}}{4}-\frac {a^{2} x^{2}}{4}-y^{3}-\frac {3 x^{2} y^{2}}{4}-\frac {3 y^{2} a x}{2}-\frac {3 x^{4} y}{16}-\frac {3 y a \,x^{3}}{4}-\frac {3 a^{2} x^{2} y}{4}-\frac {x^{6}}{64}-\frac {3 a \,x^{5}}{32}-\frac {3 x^{4} a^{2}}{16}-\frac {x^{3} a^{3}}{8}=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 41
dsolve(diff(y(x),x) = -1/2*x+1+y(x)^2+1/2*x^2*y(x)+y(x)*a*x+1/16*x^4+1/4*x^3*a+1/4*a^2*x^2+y(x)^3+3/4*x^2*y(x)^2+3/2*a*x*y(x)^2+3/16*y(x)*x^4+3/4*y(x)*a*x^3+3/4*a^2*x^2*y(x)+1/64*x^6+3/32*x^5*a+3/16*a^2*x^4+1/8*a^3*x^3,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x^{2}}{4}-\frac {a x}{2}+\operatorname {RootOf}\left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+a +2}d \textit {\_a} \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 6.348 (sec). Leaf size: 906
DSolve[y'[x] == 1 - x/2 + (a^2*x^2)/4 + (a*x^3)/4 + (a^3*x^3)/8 + x^4/16 + (3*a^2*x^4)/16 + (3*a*x^5)/32 + x^6/64 + a*x*y[x] + (x^2*y[x])/2 + (3*a^2*x^2*y[x])/4 + (3*a*x^3*y[x])/4 + (3*x^4*y[x])/16 + y[x]^2 + (3*a*x*y[x]^2)/2 + (3*x^2*y[x]^2)/4 + y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{9} \text {RootSum}\left [729 a^2 \text {$\#$1}^9+3132 a \text {$\#$1}^9+3364 \text {$\#$1}^9+2187 a^2 \text {$\#$1}^6+9396 a \text {$\#$1}^6+10092 \text {$\#$1}^6+2187 a^2 \text {$\#$1}^3+9396 a \text {$\#$1}^3+9984 \text {$\#$1}^3+729 a^2+3132 a+3364\&,\frac {729 a^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+3132 a \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+3364 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+81\ 2^{2/3} a \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^4+174\ 2^{2/3} \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^4+1458 a^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+6264 a \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+6728 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+18 \sqrt [3]{2} (27 a+58)^{2/3} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^2+81\ 2^{2/3} a \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}+174\ 2^{2/3} \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}+729 a^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )+3132 a \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )+3364 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )}{729 a^2 \text {$\#$1}^8+3132 a \text {$\#$1}^8+3364 \text {$\#$1}^8+1458 a^2 \text {$\#$1}^5+6264 a \text {$\#$1}^5+6728 \text {$\#$1}^5+729 a^2 \text {$\#$1}^2+3132 a \text {$\#$1}^2+3328 \text {$\#$1}^2}\&\right ]=\frac {(27 a+58)^{2/3} x}{9\ 2^{2/3}}+c_1,y(x)\right ] \]